A Newton algorithm for semi-discrete optimal transport with storage fees and quantitative convergence of cells

08/30/2019

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by   Mohit Bansil, et al.

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In this paper we will continue analysis of the semi-discrete optimal transport problem with storage fees, previously introduced by the authors, by proving convergence of a damped Newton algorithm for a specific choice of storage fee function, along with quantitative convergence of the associated Laguerre cells under limits of various parameters associated with the problem. A convergence result for cells in measure is proven without the additional assumption of a Poincarè-Wirtinger inequality on the source measure, while convergence in Hausdorff metric is shown when assuming such an inequality. These convergence results also yield approximations to the classical semi-discrete optimal transport problem.